The bus carrying about 30 passengers was travelling on the Indian capital's Outer Ring Road when the bag containing the bomb was spotted, the Times of India reported.
A passenger had noticed an unclaimed bag three seats behind the driver and notified the conductor who asked the bus driver, Kuldeep Singh, to stop.
"One of the passengers noticed a bag lying unclaimed near the driver's seat. When no one came forward to claim it, the driver parked the bus and ordered everyone to get out," The Times cited one passenger, named only as Narender, as saying.
"The driver saw a wire hanging out of the bag which was about two-feet long. The only solution seemed to be to throw the bag away and as the driver was doing that, the blast happened," the Sunday Express quoted the conductor, Budh Prakash, as saying.
Another witness told the Times of seeing a mad rush and a cloud of smoke as the bomb exploded.
"It was a miraculous escape"
Afzal, bus passenger
"Initially, everyone thought it was a tyre burst, but when we got closer to the spot, we realised it was much more serious," he was quoted as saying.
Both the driver and the conductor were injured in the blast. The driver was reported to be in a serious condition in hospital on Sunday while the conductor sustained minor injuries.
"It was a miraculous escape," the Times of India quoted a passenger named Afzal as saying.
The Delhi Transport Corporation (DTC) said all staff were instructed to be on the look out for any unclaimed baggage and to move people out in case anything suspicious was spotted.
"Thank God they were alert today," a DTC official said.